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Wednesday, September 20, 2017

The Legacy of Jonathan Borwein

Keith Devlin and Jonathan Borwein talk to host Robert Krulwick on stage at the World Science Festival in 2011.

At the end of this week I fly to Australia to speak and participate in the Jonathan Borwein
Commemorative Conference in Newcastle, NSW, Borwein’s home from 2009 onwards, when he
moved to the Southern hemisphere after spending most of his career at various Canadian
universities. Born in Scotland in 1951, Jonathan passed away in August last year, leaving behind
an extensive collection of mathematical results and writings, as well as a long list of service
activities to the mathematical community. [For a quick overview, read the brief obituary
written by his long-time research collaborator David Bailey in their joint blog Math Drudge. For
more details, check out his Wikipedia entry.]

Jonathan’s (I cannot call him by anything but the name I always used for him) career path and
mine crossed on a number of occasions, with both of us being highly active in mathematical
outreach activities and both of us taking an early interest in the use of computers in
mathematics. Over the years we became good friends, though we worked together on a project
only once, co-authoring an expository book on experimental mathematics, titled The Computer as Crucible, published in 2008.

Most mathematicians, myself included, would credit Jonathan as the father of experimental
mathematics as a recognized discipline. In the first chapter of our joint book, we defined
experimental mathematics as “the use of a computer to run computations—sometimes no
more than trial-and- error tests—to look for patterns, to identify particular numbers and
sequences, to gather evidence in support of specific mathematical assertions that may
themselves arise by computational means, including search.”

The goal of such work was to gather information and gain insight that would eventually give rise
to the formulation and rigorous proof of a theorem. Or rather, I should say, that was Jonathan’s
goal. He saw the computer, and computer-based technologies, as providing new tools to
formulate and prove mathematical results. And since he gets to define what “experimental
mathematics” is, that is definitive. But that is where are two interests diverged significantly.

In my case, the rapidly growing ubiquity of ever more powerful and faster computers led to an
interest in what I initially called “soft mathematics” (see my 1998 bookGoodbye Descartes) and
subsequently referred to as “mathematical thinking,” which I explored in a number of articles
and books. The idea of mathematical thinking is to use a mathematical approach, and often
mathematical notations, to gather information and gain insight about a task in a domain that
enables improved performance. [A seminal, and to my mind validating, example of that way of
working was thrust my way shortly after September 11, 2001, when I was asked to join a team
tasked with improving defense intelligence analysis.]

Note that the same phrase “gather information and gain insight” occurs in both the definition
of experimental mathematics and that of mathematical thinking. In both cases, the process is
designed to lead to a specific outcome. What differs is the nature of that outcome. (See my
2001 book InfoSense, to get the general idea of how mathematical thinking works, though I
wrote that book before my Department of Defense work, and before I adopted the term “mathematical thinking.”)

It was our two very different perspectives on the deliberative blending of mathematics and
computers that made our book The Computer as Crucible such a fascinating project for the two of us.

But that book was not the first time our research interests brought us together. In 1998, the
American Mathematical Society introduced a new section of its ten-issues- a-year Notices, sent
out to all members, called “Computers and Mathematics,” the purpose of which was both
informational and advocacy.

Though computers were originally invented by mathematicians to perform various numerical
calculations, professional mathematicians were, by and large, much slower at making use of
computers in their work and their teaching than scientists and engineers. The one exception
was the development of a number of software systems for the preparation of mathematical
manuscripts, which mathematicians took to like ducks to water.

In the case of research, mathematicians’ lack of interest in computers was perfectly
understandable—computers offered little, if any, benefit. (Jonathan was one of a very small
number of exceptions, and his approach was initially highly controversial, and occasionally
derided.) But the writing was on the wall—or rather on the computer screen—when it came to
university teaching. Computers were clearly going to have a major impact in mathematics
education.

The “Computers and Mathematics” section of the AMS Notices was intended to be a change
agent. It was originally edited by the Stanford mathematician Jon Barwise, who took care of it
from the first issue in the May/June 1988 Notices, to February 1991, and then by me until we
retired the section in December 1994. It is significant that 1988 was the year Stephen Wolfram
released his mathematical software package Mathematica. And in 1992, the first issue of the
new research journal Experimental Mathematics was published.

Over its six-and- a-half years run, the column published 59 feature articles, 19 editorial essays,
and 115 reviews of mathematical software packages — 31 features 11 editorials, and 41
reviews under Barwise, 28 features, 8 editorials, and 74 reviews under me. [The Notices
website has a complete index.] One of the feature articles published under my watch was
“Some Observations of Computer Aided Analysis,” by Jonathan Borwein and his brother Peter,
which appeared in October 1992. Editing that article was my first real introduction to
something called “experimental mathematics.” For the majority of mathematicians, reading it
was their introduction.

From then on, it was clear to both of us that our view of “doing mathematics” had one feature
in common: we both believed that for some problems it could be productive to engage in
mathematical work that involved significant interaction with a computer. Neither of us was by
any means the first to recognize that. We may, however, have been among the first to conceive
of such activity as constituting a discipline in its own right, and each to erect a shingle to
advertise what we were doing. In Jonathan’s case, he was advancing mathematical knowledge;
for me it was about utilizing mathematical thinking to improve how we handle messy, real-world problems. In both cases, we were engaging in mental work that could not have been
done before powerful, networked computers became available.

It’s hard to adjust to Jonathan no longer being among us. But his legacy will long outlast us all. I
am looking forward to re-living much of that legacy in Australia in a few days time.

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The Mathematical Association of America is the world's largest community of mathematicians, students, and enthusiasts. We accelerate the understanding of our world through mathematics, because mathematics drives society and shapes our lives. Visit us at maa.org.